Alcatos of Stochastc Models o Test ms-scalg Educatoal ad Psychologcal Measuremet Dr. R.Arumugam, M. Raja Assstat Professor,Deartmet of Maematcs, Peryar Maamma Isttute of Scece ad Techology, Thajavur- 63403. Assstat Professor,Deartmet of Educato, Peryar Maamma Isttute of Scece ad Techology, Thajavur -63403 Abstract : I s aer stochastc models are develoed to study e rate of test ms-scalg educatoal ad sychologcal measuremet. The mea ad varace of geometrc ad egatve bomal models are gve to esure e umber of ms-scalg below a reshold. Numercal llustrato are also rovded. Keywords: Stochastc model, Ms-scalg ad Threshold. I. INTRODUCTION I e last decade, essay tems have bee cluded major educatoal assessmets, such as e Natoal Assessmet of Educatoal Progress (NAEP) ad e Thrd Iteratoal Maematcs ad Scece Study (TIMSS) (Alle, Carlso, & Zelea, 999; Mart & Kelly, 996). Meawhle, classroom teachers are advsed to use essay questos to comlemet multle-choce tems. Varous resoses geerated from essay tems demad a large amout of maower test gradg. Whle o graders la to mae mstaes, accdetal errors are lely to occur durg e huma oeratos (Wag, 993). The urose of s study s to exame e chace of test ms-scalg usg arorate models statstcs. The estmato of advertet gradg errors ca serve as a bass for qualty cotrol educatoal ad sychologcal measuremets. II. LITERATURE REVIEW Statstcal models have bee sought to ehace qualty cotrol varous rojects. I a dustral statstcs, qualty cotrol measures are adoted maly to esure e total umber of feror cdets below a reshold. Bssssell (970) revewed; It s ofte assumed at such evets follow e Posso Law. The assumtos of costat mea level ad deedece are ofte volated ractce (. 5). I educatoal ad sychologcal measuremets, test ms-scalg ca be treated as a secfc d of cdets. I a classroom settg, Lyma (998) oted at "Every teacher recogzes at grades are raer arbtrary ad subjectve" (. 07). I a large-scale assessmet, t s eve more dffcult to assume e same level of average erformace amog varous graders. Hece, e assumto of a costat mea erformace level s ofte volated large ad small-scale assessmets, whch maes e Posso model arorate for most real-lfe alcatos (Rasch, 980; Wag, 993). III. STOCHASTIC MODELS I a test scorg rocess, a cotrast ca be set to dfferetate outcomes of correct gradg ad ms-scalg. A evet w dchotomous outcomes s tycally modeled by a Beroull tral. For a welldesged test, e ossblty of ms-scalg () s ot hgh. Qualty cotrol measures, such as arragemet of schedules for short breag, ca be troduced e gradg rocess to esure at e DOI:0.3883/IJRTER.08.438.6DDO 33
Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: 455-457] umber of ms-scalg cases s o larger a a secfc level. I ractce, e gradg rocess may ee o utl occurrece of e ms-scalg. The, a brea sesso ca be scheduled to refresh e graders, ad us, hel cotrol e umber of ms-scalg below level. To facltate descrto of e stochastc model, oe may defe X to be e total umber of successes before e ms-scalg, ad to be e robablty of obtag exactly X successes. Accordgly, e total umber of trals deeds o e reshold level ad e umber of correctly-graded cases (X) before reachg e reshold. Sce e evet of ms-scalg haes by accdet, e umber of correctly-graded cases (X) may vary amog e graders. Gve a level of e reshold, e exected value of X ca be emloyed to schedule brea sessos before reachg a ms-scalg cdet o e tral. f (x) ( X ) ( X ) IV. GEOMETRIC STOCHASTIC PROCESS Uder a codto of zero tolerace, oe may wsh to schedule a brea erod for test graders before e frst occurrece of ms-scalg. Usg symbol s to rereset successful test gradg ad m to rereset e frst ms-scalg, oe may dect e stochastc rocess e followg cha of evets: sssm X tm e s Ths stochastc cha ca arse oly oe way,.e., e grader has successfully graded test questos o e frst X trals, ad eds u w e frst ms-scalg case o e ( X ) tral. Thus, e chaces of obtag exactly X successes ror to e frst falure s: ( ) ( )...( ) Where, s e robablty of ms-scalg each tral. Ths stochastc rocess ca be descrbed by a robablty fucto for dfferet values of X: x ( ) ; x 0,,,,3,......( ) Equato () defes a geometrc rocess "because e robabltes form a geometrc seres w a commo rato ". Sce e ossblty of ms-scalg each tral () s less a, e total robablty follows: ( ) ( )... ( ) V. MAIN RESULTS The exected umber of correctly graded s, X)............ () Sce a brea sesso s arraged before evet of e frst ms-scalg (.e., =), e watg tme for e frst ms-scalg s: X ) X ) X)......... (3) Sce, X ), X ) V( X ) @IJRTER-08, All Rghts Reserved 34
Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: 455-457] Table : The relatosh betwee P ad x) 0. 0. 0.5 0.8 0. 0.3 x) 0.00 8.33 6.67 5.55 5.00 3.33 Table : The relatosh betwee P ad V(x) 0.0 0. 0.5 0.8 0. 0.3 V(x) 90.00 6. 37.78 5.3 0.00 7.78 For a gve test, f e chace of ms-scalg () s small, e e watg tme for a brea sesso ca be loger. I a extreme case, multle-choce (MCQ) tests are graded by a mache whch has equal to zero each tral ad e watg tme ca be fte. Therefore, ere s o eed for a brea uless e gradg mache s broe dow. VI. A NEGATIVE BINOMIAL MODEL I a more comlcated codto, a test s graded by a total umber of graders. For ay grader, w reshold deed o e qualty cotrol requremet, ad may tae a value larger a oe. To mae sure e ms-scalg evet occurrg o e ( X ) tral, e revous ( ) ms-scalg evets must already hae e recedg ( X ) trals. The robablty of ( ) ms-scalg cases o e frst X ) trals s gve by ( X X ( ) The robablty of ms-scalg o tral X ) s. ( X 0,,,...... (4) @IJRTER-08, All Rghts Reserved 35
Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: 455-457] Equato (4) follows a egatve bomal dstrbuto (Feller, 957). The robablty geerato fucto for e egatve bomal dstrbuto s ( ) t. Because e overall qualty cotrol s based o cumulatve erformace of e graders, e stochastc rocess volves deedet radom varables,, X X. Thus, e qualty cotrol reshold hges o e dstrbuto of X X,... Lucly, e robablty geerato fucto P X ( t) X follows ( ) t ( ) t.........(5) Where,.... Based o uqueess of e robablty geeratg fucto (Port, 994), must have a egatve bomal dstrbuto w arameters ad. X) The exected value for ( ) ad V( X ca be resultg from e egatve bomal dstrbuto, ad e result s (see Casella & Berger, 990). So, e watg tme before occurrece of e ( ) X ) q X )............ (6)............(7) Table 3 : The relatosh amod, ad x) 0.5 0.4 0.5 0.3 0.45 0.45 0.54 0.54 x) 7 4 4 3 ms-scalg s:. Table 4 : The relatosh amod,, q ad V(x) 0.5 0.4 0.5 0.3 0.45 0.48 0.54 0.59 q 0.85 0.76 0.75 0.70 0.55 0.5 0.46 0.4 V(x) 6 3 3 3 @IJRTER-08, All Rghts Reserved 36
Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: 455-457] VII. DISCUSSION I a comarso betwee (3) ad (6), oe may ote at e geometrc rocess ca be treated as a secal case ( = ) of e egatve bomal dstrbuto. Based o e results (6), e watg tme for a brea erod ca be loger f e overall tolerace level s hgher ad e chace of ms-scalg () s small. I summary, artly due to dffereces e otato choce, e well-establshed geometrc ad egatve bomal dstrbutos have yet to be used models of test ms-scalg. I oer felds, Johso ad Kotz (969) have oted at "e egatve bomal dstrbuto s frequetly used as a substtute for e Posso dstrbuto whe t s doubtful wheer e strct requremets, artcularly deedece, for a Posso dstrbuto wll be satsfed" (. 35). Thus, e geometrc ad egatve bomal models rovde alteratve choces at are more flexble a e Posso model educatoal ad sychologcal measuremets. VIII. CONCLUSION The geometrc rocess s develoed from a sgle-grader scearo uder a olcy of zero tolerace for test ms-scalg. Hece, e result equato (3) may be more alcable a local settg whch a teacher has bee assged to grade tests for a etre class. The egatve bomal rocess, o e oer had, seems arorate for state or atoal assessmet at volves more a oe test grader. I bo cases, e watg tme for test ms-scalg has bee derved from e corresodg stochastc rocesses. The results ca be emloyed to schedule brea erods to esure e error of msscalg below a reshold. REFERENCES I. Alle, N. L., Carlso, J. E., & Zelea, C. A. (999). The NAEP 996 techcal reort.washgto, DC: Natoal Ceter for Educato Statstcs. II. Bssell, A. F. (970). Aalyss of data based o cdet couts. The Statstca, 9 (3), 5-47. III. Casella, G ad Berger, R. L. (990). Statstcal ferece. Pacfc Grove, CA: Broos. Draer, N. R., & Lawrece, IV. W. E. (970). Probablty: A troductory course. Chcago, IL: Marham. Feller, W. (957). A troducto to robablty eory ad ts alcatos (d ed.). New Yor, NY: Joh Wley & Sos. V. Lyma, H. B. (998). Test scores ad what ey mea. Bosto, MA: Ally & Baco. VI. Rasch, G. (980). Probablstc models for some tellgece ad attamet tests.chcago, IL: Uversty of Chcago Press. VII. Wag, J. (993). Smle ad herarchcal model for test msgradg. Educatoal ad Psychologcal Measuremet, 53, 597-603. @IJRTER-08, All Rghts Reserved 37